Difference between revisions of "Delta integral of certain shift of f is delta integral of f"
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(Created page with "==Theorem== The following formula holds: $$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ where $\hat{f}$ denotes th...") |
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The following formula holds: | The following formula holds: | ||
$$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ | $$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ | ||
| − | where $\hat{f}$ denotes the solution of the [[shifting problem]]. | + | where $\displaystyle\int$ denotes the [[delta integral]], $\sigma$ denotes the [[forward shift]], and $\hat{f}$ denotes the solution of the [[shifting problem]]. |
==Proof== | ==Proof== | ||
Revision as of 14:34, 21 January 2023
Theorem
The following formula holds: $$\displaystyle\int_{t_0}^t \hat{f}(t,\sigma(\xi))\Delta \xi=\displaystyle\int_{t_0}^t f(\xi) \Delta \xi,$$ where $\displaystyle\int$ denotes the delta integral, $\sigma$ denotes the forward shift, and $\hat{f}$ denotes the solution of the shifting problem.
